The mathematics of perfect shuffles

ADVANCES

IN APPLIED

4, 175- 196 ( 1983)

MATHEMATICS

The Mathematics

of Perfect Shuffles

PERSI DIACONIS Stanford

University,

Stanford

California 94305 and Harvard Massachusetts 02138

Universiv,

Cambridge,

R. L. GRAHAM Bell Laboratories,

Murray

Hill,

New Jersey 07974 and Stanford California 94305

WILLIAM University

of Oregon,

University,

Stanford,

M. KANTOR

Eugene, Oregon 97403 and Bell Laboratories, New Jersey 07974

Murray

Hill,

There are two ways to perfectly shuffle a deck of 2n cards. Both methods cut the deck in half and interlace perfectly. The out shuffle 0 leaves the original top card on top. The in shuffle I leaves the original top card second from the top. Applications to the design of computer networks and card tricks are reviewed. The main result is the determination of the group (I, 0) generated by the two shuffles, for all n. If 2 n is not a power of 2, and if 2n * 12,24, then (I, 0) has index 1,2, or 4 in the Weyl group B, (the group of all 2”n! signed n x n permutation matrices). If 2n = 2“, then (I, 0) is isomorphic to a semi-direct product of Zi and Z,. When 2 n = 24, (I, 0) is isomorphic to a semi-direct product of 2-j’ and M,,, the Mathieu group of degree 12. When 2n = 12, (I, 0) is isomorphic to a semi-direct product of Zi and the group PGL(2,5) of all linear fractional transformations over GF(5).

1. I~RODUCTI~N

There are two ways to perfectly shuffle a deck of 2n cards. Both cut the deck in half and interlace perfectly. The in shuffle I leaves the original top card second from the top. The out shuffle 0 leaves the original top card on top. Let the deck be labeled (0, 1,. . . , n - 1, n,. . . , 2n - 1). After an in shuffle the order is (n, 0, n + 1,. . . , 2n - 1, n - 1). After an out shuffle, the order is (O,n,l,n+ l,..., n - 1,2n - 1). These shuffles have been used by gamblers and magicians to manipulate cards. A historical review, 175 0196-8858/83 $7.50 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.

176

DIACONIS,

GRAHAM,

AND

KANTOR

describing “widely known” properties of the shuffles, is in Section 3. This section also describes results of Levy on the cycle structure of the shuffles, and results concerning decks of odd size. In and out shuffles appear in computer science as a way of connecting processors in parallel processing machines. One widely known application is an O(log n) FFT algorithm. Section 4 discusses these applications as well as some new results; for example, an array of 2k numbers can be reversed in k in shuffles. Section 5 discusses some related permutations: Levy’s work on the “milk shuffle” and Morris’ work on generalized perfect shuffles. The main result of this paper is a determination of the group generated by in and out shuffles. This group will be called the shuffle group and denoted (I, 0). Both of the generators I and 0 preserve central symmetry; that is, cards symmetrically located about the center of the deck (0 and 2n - 1, 1 and 2n - 2, etc.) are sent to positions symmetric about the center. Thus (I, 0) is a subgroup of the centrally symmetric permutations. This group is isomorphic to the Weyl group B,, of n X n signed permutation matrices. This is the group of all n x n matrices with entries 0, f 1, and one nonzero entry in every row and column. Equivalently, it is the group of all 2”n! symmetries of the n-dimensional generalization of the octahedron, whose vertices are fe,,. . . , f e,, where e,,. . . , e, is the standard basis of BP” (see Coxeter [5]). The pairs {ei, - ei} correspond to the pairs of card positions which are centrally symmetric. We will have to deal with three homomorphisms ofB,,. If g E B,, then sgn(g) is its sign as a permutation of 2n cards, and sgn(g) is its sign as a permutation of n centrally symmetric pairs; and g + sgn(g)g(g) is a further homomorphism to { f l}, whose kernel is the Weyl group 0,. THEOREM. Let (I, 0) be the permutation shuffles of 2n cards.

group generated by in and out

an d n > 6, then (I, 0) is isomorphic to B, and n = 6, then (I, 0) is a semi-direct product of Z,6 with I(Z, 0)l = n!2”. If PGL(2,5). (b) If n = 1 (mod4) and n > 5, then (I, 0) is the kernel of s@ and I(I, 0)l = n!2”-‘.

(a) If n = 2 (mod4)

(c) If n = 3 (mod4), then (I, 0) is isomorphic to D,, and I(I, O)l = . (4 If n = 0 (mod4), n > 12, and n not a power of 2, then (I, 0) is the intersection of the kernels of sgn and sz, and I( I, O)l = n!2”- 2. If 2n = 24, then (I, 0) is a semi-direct product of Zi’ with the Mathieu group M,, of degree 12. (e) If 2n = 2k, (I, 0) is. isomorphic to the semi-direct product of Z,k by Z,, where Z, acts by a cyclic shift and I(I, O)l = k * 2k. n!2

n-1

PERFECT

177

SHUFFLES

TABLE I Order of the Group (I, 0); M = 2”n ! 2

2n

4

I(I,O)(

2

2n

20

22

I([* O)l

M

M/2

2n

38

40

I(I~O)l

M/2

M/4

6

2 . 22

8

10

3. 23

12

14

16

M/2

M/3!

M/2

26

28

30

32

M/2

M

M/2

42

44

46

48

50

52

M/2

M

M/2

M/4

M/2

M

M/2 24 M/(7!

. 2)

5

4. 24

25

18 M/2

34

36

M/2

M

The theorem, and a number of other results about the shuffle group, are proved in Section 2. A list of the order of the shuffle group for 2n = 2,4,. . . ) 52, appears in Table I. This table was computed using a streamlined implementation of an algorithm of Sims [24] developed by Eric Hamilton and Donald Knuth at Stanford University. The numerical evidence allowed us to guess at the theorem. For a discussion of Sims’ algorithm see Furst et al. [7]. Finally, we mention the connection between our shuffles (for a deck of 24 cards) and some very recent work of Borcherds et al. [2] on representations of the Leech lattice in hyperbolic space. Section 5 contains further discussion. 2.

PROOF

OF

THE

THEOREM

We begin by establishing a number of basic properties of perfect shuffles. Lemma 1 gives the order of in and out shuffles. It is well known: see Uspensky and Heaslet [26, pp. 244-2451, Herstein and Kaplansky [ll, Chap. 3.41. LEMMA

(mod2n

1. The order of the in shuffle permutation is the order of 2 + 1). The order of the out shuffle is the order of 2 (mod2n - 1).

Proof The order of an in shuffle is the order of an out shuffle with a deck containing 2 more cards, so we only prove the result for out shuffles. If the deck is labeled 0 , 1,***, 2n - 1, then after one out shuffle the card labeled j is at position 2j (mod 2n - 1) if j < 2n - 1. After k shuffles it is in position 2&j (mod 2n - 1). All cards will be in their original positions for the smallest k such that 2k = 1 (mod2n - 1). 0 Remark. A pack of 52 cards requires 8 out shuffles or 52 in shuffles to recycle. Because of Fermat’s theorem, the pack will always recycle after

178

DIACONIS,

GRAHAM,

AND

KANTOR

at most 2n - 2 out shuffles, although fewer may do. A famous conjecture of Artin asserts that 2 is a primitive root (mod p) for infinitely many primes p. If this is true, there are arbitrarily large n such that 2n - 2 out shuffles are required to recycle 2n cards. Lemma 2 gives an algorithm for bringing the top card to any position by a sequence of in and out shuffles. The result was first given by Alex Elmsley. For a proof, see Morris [21, Proposition 11. LEMMA 2. Let the positions in a deck of 2n cara% be labeled 0, 1, . . . , 2n - 1. To bring the top card to position k, expressk in binary, interpret 1 as in and 0 as out, and perform the indicated sequenceof in and out shufflesfrom left to right. Lemmas 1 and 2 give the following lower bound for the shuffle group: LEMMA 3. The order of the shuffle group is at least 2n X order of 2 (mod2n - 1). The next result proves part (d) of the theorem. It establishes that the lower bound of Lemma 3 is achieved for decks of size 2’. The proof provides a simple interpretation of in and out shuffles as maps on the vector space Zf. LEMMA 4. For 2n = 2k, the shuffle group is isomorphic to a semi-direct product of Zt by zk. Proof. Label the cards using their binary expansions (xt,. . . , xk). It is easy to check that, using an obvious notation,

0: (x+1 ,..., xk) + (xz,x~,-, xk,x,), 1: (x~,x~,...,xk)-,(x~,x~,...,xk,x,)

withx,

= 1 - x,.

The permutations Oj, 0 Q j < k, form a cyclic group (0). The permutations Bj = Oj-‘10-i send (x,,. . . , xi,. . . , xk) into (x,,. . . , Xj,. . ., xk), for 1 gjgk.MapZ,kintotheshufflegroupbysendinga=(a,,...,a,)~Z,k into l-l,“,,B,?~. It is easy to see this is an isomorphism. The image group intersects the cyclic group (0) only in the identity and their product contains 0 and I. The action of Z, on Zt is a cyclic shift. 0 Both in and out shuffles preserve arrangements with “central symmetry.” We now turn to definitions and amplification of this fact. Throughout, permutations will be applied on the right, so xg is the image of x under the permutation g. S,, and A, denote the symmetric and alternating groups of degree n. If S is a set of permutations, then (S) denotes the group generated by S.

179

PERFECT SHUFFLES

If a < b, then [a, b) = (x E Zla d x < b}. A permutation g E S,, has central symmetry if (i)g + (2n - 1 - i)g = 2n - 1 for all i E [0, n). The subgroup of centrally symmetric permutations is denoted B,,. It is natural to label the objects being permuted as 0, 1, . . . , n - 1, (n l)‘, . . . ) l’, 0’, where x f) x’ is the natural pairing. Let X = {x, x’}. A permutation g E B,, induces a permutation g of {X]x E [0, n)}. The map g + g is a homomorphism from B,, onto S,, with kernel equal to ((0,O’)) X ((1,1’)) X ... x ((n - l,(n - 1)‘)) E Z;. Note that G(g) = sgn(g). DEFINITIONS.

In this notation the shuffles can be written 0:

I:

x + 2x

i x + (2(n - x) - 1)’ x+2x+1 i x + (2(n - x - 1))’

and x’ --) (2x)‘,

if x E [0, fn),

and

x’+2(n-x)-l,

ifxE[fn,n);

and

x’ + (2x + l)‘,

ifx E [O,(n

and

x’ + 2(r1 - x - l),

if x E [(rz - 1)/2, r~).

- 1)/2),

Let G = (I, 0). Let c be the image of G in S,, and let K be the kernel of G + c. LEMMA

5.

When n = 12, c is isomorphic to M,, and K is isomorphic to

Z;‘. Proof: The group c is generated by 0 = (i%@??sm%fl) and 7 = ---------(0137861025 11)(49). A computer check shows that the order of cis 95,040, _. the order of M,2. Further, G’ is doubly transitive; indeed g is an 1 l-cycle fixing 0 and Z moves 0, so G is transitive, and to bring symbols labeled (i, j) to postions (k, I), bring i to 0, bring j to the appropriate place by iterates of 0, and then bring i to k. Sims [24] showed that M,, is the only doubly transitive permutation group of this order. The result for K follows from the order of G given in Table I. 0

We were intrigued by the appearance of M,,. Table II lists 66 6-sets; these sets and their complements in [0,12) form the design S(5,6,12): each 5-set is in exactly one of the 132 6-sets. Moreover, Aut S(5,6,12) = M,,. Remark. Lemmas 4 and 5 are concerned with two of the three exceptional situations appearing in the theorem. The third situation occurs when 2n = 12, and is less interesting. In this case, (X]X E [0,6)} will be identified with GF(5) u {co} as follows:

Then 0 sends x + x + 2 and f sends x + 2/x.

This identifies

c with

PGL(2,5). Since Z605 = (O,O’), the transitivity of G yields that /Kl = 26. If k = (2,2’) and I = (3,3’), then k -‘02k and I -‘Z41 send [0,6) to itself and

180

DIACONIS,

GRAHAM,

AND

KANTOR

TABLE II A Design with Automorphism 1

5

2

10

711 9

1

0

4:

3

0

8

617025

11

8

0

Group M,,

1

9

0

6

011

11

1

12

7

8

9

7

I

435207

11 0

9

0

41040

7

11

3

8

0

5

2

8

10

0

7

9

1

6

7

0

10

4

7

3

0

911

211

4

9

6

5

10

2

8

145083 6

9

0

3

896051

4

1

5

0

6

7

5

11

0

10

2

3

8

2

10

0

11

9

10

1

0

3

4

11

9

532068 8

6

0

2

10

6

4

7

6

5

0

10

3

5

7

110

9

11

10

0

3

6

10

‘9

2

0

611

5

13

1L

5

0

4

8

7

3

10

7

8

0:2

7

0

0

7

9

5

11

6

0

9

5

10

1

11

0

5

10

3

2

1

0

10

3

6

4

2

0

3

6

11

8

4

1 10

0

3

8

6

6

3

0

4

6

011

8

11

0

1

8

11

5

0

7

3

7

1

10

0

920438

5

4

6

0

10

8

11

0

11

0

1

2

2

0

4

8

211

9

9

10

10

0

11

1

5

8

6

6

4

681024

4

4

11

2

3

7

3

11

354690

10

11

0

2

2

087963

10

674301

1 0

169230

1

1

5

2

0

2

10

4

0

7

11

9

540867 10

8

4

0

8

7

4

823710

6

9

0

5

2

10

746920

11

5

0

10

4

3

258079 4

10

7

0

9

5

8

3

9

0

5

10

9

1

711 9

811

5

4

370915

0

induce PGL(2,5) there. Consequently, (I, 0) is a semi-direct asserted in the theorem. (N.B. - PGL(2,5) = S,.)

product as

LEMMA 6. (aJIJ n = 0 (mod 4), then G is in the kernels of the homomorphisms sgn and sgn.

(b) If n = 1 (mod4), then G is in the kernel of &$. (c) Zf n = 3 (mod4), then G is in the kernel of sgn . G. Note that parity considerations give no restrictions on G when n = 2 (mod4). On the other hand, all homomorphisms of B, in the lemma are onto {f l}, so that the orders of the groups given in the theorem are upper bounds.

PERFECT

181

SHUFFLES

Proof. It suffices to prove that the signs of Z, Z, 0, and Dare as in Table III. Since the case of Z follows from that of 0 by replacing n by n + 1, we only need to consider the parity of 0 and ??. In order to deal with 0, we label the cards 0, 1,. . . , 2n - 1, so x0 = 2x (mod 2n - 1). Let x -Cy. We need to decide when x0 > y0. Since x0 = 2x or 2x - (2n - 1) and x c y, the inequality x0 > y0 holds if and only if 0 < x d n - 1, n < y < 2n - 1, and 2x > 2y - (2n - 1). Thus, for each x E [0, n - 1) we must restrict y to [n, x + n). The number of pairs (x, y) is then c::ix = 0y . Next,recallthata=%or2(n-x)-l.LetXg. There are two ways this can happen:

(i) x, y > n/2, in which case 2(n - x) - 1 > 2(n - y) - 1 holds, or (ii) x < n/2, y 2 n/2, and 2x > 2(n - y) - 1, that is, x + y >, n. Set m = [(n + 1)/2]. The number of (js, 7) in (i) is (n 2 m). The number in (ii) is obtained by fixing x E [0, m), and then letting y E [n - x, n). Thus, the total number of pairs (x, 7) is

If n is even this is 2 ( T 1; if n is odd it is (n - 1)2/4. It is now straightforward to check Table III and then deduce Lemma 6. 0 Lemma 7 will be used several times. The simple proof is omitted. LEMMA 7. Suppose n >, 3. Form a graph with vertices the 3-cycles in A,,. Join two 3-cycles when some point of [0, n) is displaced by both of them (i.e., when the corresponding 3-sets are not disjoint). Let H be a subgroup of A,, generated by a connected set of 3-cycles. Zf each point of [0, n) is displaced by a 3-cycle in H, then H = A,. We now begin the main part of the proof of the theorem. Some further

notation will be required. If g and h are permutations,

(w(g),

TABLE III w(g)) for g = Z or 0 n (mod 4)

1

and in cycle notation

182

DIACONIS,

GRAHAM,

AND

KANTOR

his se-( . . . . x,y ,... )..e, thenhs=g-‘hgis me-( . . . . xg,yg ,... ).... Let z be the element of S,, interchanging x and x’ for all x E [0, n); thus, z has the effect of reversing the order of the deck. Let G* consist of the elements of G sending [0, n) to itself. LEMMA 8. If G* 2 A, then IK( a 2”-‘. - -. By Table III, at least one of the permutations I, 0 is even. Let h E {I, 0} with 6 even. Let f E G* with f = K, and set k = f - ‘h. Then k E K - (z), k fixes 0 and o_l, and k-interchanges certain pairs x, x’. Pick any g E G* such that @ = x, G = 0 for some such pair x, x’, and such that z = r whenever yk = y. Then kg = (x)(x’)(OO’) * . . and kkg = (OO’)(xx’). Conjugating by elements of G* produces all permutations Prooj

(aa’)(bb’).

Thus pq > 2”-‘.

0

The argument is easiest for odd n. By Table I we may assume that n > 3. LEMMA 9. Proof:

The theorem ho& if n is odd.

We calculate the following permutations: IO-‘:

x + (n - 1 - x)’

I-,o:

1, I n - 1 --, (n - 1)’ 2x*2x+

1 2x + (n - 2 - 2x)‘,

I-‘O-IO-‘:

2x+l+(n-1-2x)‘, n-1+0;

if x c (n - 1)/2 (pairwise adjacent transpositions); if if

x < (n - 1)/2, x < (n - 1)/2,

;,:l-+-;;~‘l , 2x + 2x + 2,

b:= (I-‘OIO-‘)2:

(a cut at the center);

if

x < (n - 1)/2,

if

0 < x < (n - 1)/2,

1 + 0’.

Then b = (0,2,4 ,...,

n - 3, n - 1, (n - 2)‘, (n - 4)‘,. . . ) l’)(O’, 2’,. . . ) l),

c:= b”-’ = (0,1,2 ,..., c2 = (0,2,4 ,...,

(cy’O=

n - l)(O’,l’,...,

(PJ - 1)‘),

n - 3, n - 1,1,3 )...) n - 4, n - 2)(0’,2’)...)

(1,3,5 )...) n - 2,(n - 1)‘,0,2 )..., n - 5,n - 3) x (1’,3’)...)

(n - 3)‘),

(Fr - 2)‘),

PERFECT

i):=

c-yc2)~-‘~

=

, n - l,l)(O,

(0

UC-’ = ((n - l)‘, n - 2,o)(n w:= (Dc-‘)l-‘o

183

SHUFFLES

- l,(n

(n - l)‘, l’), - 2)‘,0’),

= (n - 1 ) n - 3,l)((n

- l)‘,(n

- 3)‘, 1’).

Clearly, c and w belong to G*. Restricting G* to [0, n) it is easy to check the connectedness of its set of 3-cycles. Thus, Lemma 7 yields that G* 2 A,. By Lemma 8, (K] 2 2”-‘. If n = 3 (mod4), then Gcontains A,, and the odd permutation a, so that G = S,,. Since z is an odd element of S,,, z @ G by Lemma 6(c). Thus 1K 1 = 2” - ‘, completing the proof in this case. If n = 1 (mod4), then c = A, by Lemma 6(b). Thus, there is a g E G* with g = 1 Clearly, g is even, while I is odd by Table III. Now g1- ’ is odd and belongs to K. Since K already contains all permutations of the form (xx’)( ~JJ’) (see the proof of Lemma 8), it follows that ]K] = 2”. Then G consists of all permutations in B, that map to even permutations of S,. 0 The remainder of this section is devoted to the proof of the theorem when n is

euen. This will be accomplished

in a sequence of lemmas.

Notation. 2n = 2% with 1) > 1 and 2) odd. (The manner in which o > 1 is used can be seen in Lemmas 10 and 12.) Throughout the proof, r will belong to [l, k) (except that r E [l, k] in Lemma 11).

Lemma 10 determines the result of 0 - ‘I’. In the language of cards, the effect of this sequence of shuffles is to reverse the top 2’ cards, and each consecutive group of 2’ cards, in place. LEMMA 10. If O~a 1 and O-r+‘Ir-’ behaves as indicated. We must distinguish between the cases p = 2y and p = 2y - 1. If /3 = 2y, then 2’-‘a + (2r-1 - 1 - y) < 2’-‘u Q in, and (2'a + #qo-'o-

r+lIr-lI

=

@-la

= {2'-'a = 2{2'-'a

+

y)o-r+lp~

+ (2r-1 + (2r-1

- 1 - y)}l

- 1 - y)>( + 1

= 2'a + 2' - 1 - /3.

184

DIACONIS,

Similarly,

GRAHAM,

AND

KANTOR

if j3 = 2y - 1, then 2’a + /I is odd, 2’-‘(a

+ 1) G in, and

(2’a + /3)0-‘O-‘+‘I’-‘I =

2’a

n-

+

P

+

1

2

1

‘o-r+lIr-lI

)

= {n - 2’-‘(a

+ 1) + (2’-’

= {n - 2’-‘(a

+ 1) + 2’-’

= 2n - 2 - 2{n - 2’-‘(a

- y)}‘O-‘+‘I’-‘I - 1 - (2’-’

- y)}‘l

+ 1) + y - l}

= 2’(a + 1) - 2y = 2’a + 2’ - 1 - j3. 0 Lemma 11 determines the result of I’0 - ‘. In the language of cards, the effect of this sequence of shuffles can be described as follows: with the deck on the table, cut off the top n/2’-’ cards and place them on the table. Cut off the next n/2’-’ cards and place them on the original top group. Continue cutting off packets of size n/2’-‘, placing them on the cards already cut off, until there are no more cards left. LEMMA 11. If r E [l, k] and x E ((i - l)(n/2’-‘) with 1 < i Q 2’-‘, then

xI’O--‘= 1-x-l+ a&x’I’O-’

Proof:

=

-X

- 1 + -32i

+2i

- 1, i(n/2’-‘)

- l]

- 1))’

- 1).

The proof is again inductive. If r = 1 the lemma states that xIO-’

= {-x

- 1 + n}’

for

x E(-1,

n - 11.

Let r 3 2. We must distinguish between the cases x E [0, fn) x E [in, n). Let x E [0, fn). Then XI = 2x+ 1 E ((i-l)(n/2’-2)-1, i(n/2’-2)and XII’ - ‘0 - r+‘o-’

=

(

- (2x + 1) - 1 + -32’

= z1 -2x-2+” ( [

2r-2

(2’ 1

- 1))‘0-1

- o]) I.

and 11

PERFECT

185

SHUFFLES

Next, let x E [in, n), so that xl = (2n - 2x - 2)’ and -i-+

+ 1 d - (2x + 1) < - (i - 1)s 2n-in

(2’-’

d x’I < 2n - (i - l)- 2r-2’

2r-2

- i)*

+ 1,

- 1 < x’I < (2’-’

- i + 1)--11 2r-2

- 1.

Thus, xII-‘+‘o’-‘o-’ = = t = t

- (2n - 2x - 2) - 1 + -+{2(2r-1

- i + 1) - l}]O-’

n - + - (2n - 2x - 2) + L(2’ 2r-2

- 2i + I)]) ’

[

-x-1+-

: --&(2i

LEMMA 12. Set b = (IkOek

- 1))‘. 0 * I-‘O)-2.

Then b induces

2u - 2)(2u - 1,2u - 3,.. .) 3,1)

(WA...,

on [0,2u), and (x + 2u)b = xb + 2u whenever x, x + 2u E [0, n). In purtitular, b has order u. Proof:

Recall that u = n/2k-‘. x’IkOek

Let x E [(i - l)u, iv). By Lemma

11

= -x - 1 + u(2i - 1) E [(i - l)u, iu).

If x is even, then x’IkOpk.

I-‘0

= -x + u(2i - 1).

If x is odd, then x’IkO - k . I-‘0

= -x - 2 + u(2i - 1).

Thus, x’I~O-~I’O E [(i - l)u, iu) unless -x + u(2i - 1) = iv and x is even, or -x - 2 + u(2i - 1) = (i - 1)~ - 1 and x is odd. Excluding these cases, we find that xb -’ = - {-x

+ u(2i - 1)) - 2 + u(2i - 1) = x - 2

if x

is even,

xb -I = - {-x

- 2 + u(2i - 1)) + u(2i - 1) = x + 2

ifx

isodd.

186

DIACONIS,

GRAHAM,

AND

KANTOR

Finally, if x = (i - 1)~ is even, then xb -’ = - {-x

+ u(2i - 1)) - 2 + u(2i + 1) = x + (2u - 2)

while if x = iv - 1 is odd, then xb-’

= - {-x

- 2 + u(2i - 1)) + u(2i - 3) = x - (20 - 2). 0

LEMMA 13. Set c: = born’. Then c induces (0,1,2,. and (x + u)c = xc + u wherzeuer x, x + u E [0, n).

. . , u - 1) on [0, u),

Proofi Since b induces (2iu, 2iu + 2,. . . , 2iu + 2u - 2) on the even integers in [2iu, 2iu + 2u), c acts as indicated on [0, in). Also

((2iu + 2u - l)‘, (2iu + 20 - 3)‘,. . . , (2iu + 3)‘, (2iu + l)‘)O-’ = (n-iv-u,n-iu-u+ l,...,n-iu-2,n-iu1). 0 Notation. h(r):

forr E [l, k),

= O-1’

h: = b(l), H = H(1): = (b, c, h) H(r):

- l), h(r))

= (H(r

LEMMA 14. If g E H(r), [0,2’u)g

withbandcasinLemmas forr E (1, k).

then

= [,0,2’u)

and

whenever x, x + 2’ E [0, n). In particular, Proof.

12and 13,

This follows immediately

(x + 2’u)g = xg + 2’0 H(r)

c G*.

from Lemmas 10, 12, and 13. 0

LEMMA 15. If u > 3, then H induces S,, on [0,2u). Proof:

We will write elements of H as permutations b = (0,2,4 ,...,

c=(o,1,2

of [0,2u). Then

2u - 2)(2u - 1,. . .) 3, l),

,...) u - l)(u, 0 + 1,. . .) 2u - l),

h = (01)(23)...(u

- 3,u - 2)(u - l,u)(u

X (2~ - 4,2u - 3)(2u - 2,2u - 1). Note that h is an odd permutation (of [0,2u)). We will exhibit a 3-cycle. We have l,O)(l,u+2)(~+3,~+4)... x (2u - 2,2u - l)(u, 0 + 1)

hc2=(23)...

(u-

+ l,u + 2)...

PERFECT

187

SHUFFLES

so that (since o & 5) d:= hh” = (0, u + 2, u)(l,

2, - 1, B + 1).

Then d’ = (1, u + 3, u + 1)(2,0, u + 2), e:= ddc = (u + 2,2, u)(u

f:=

eb = (u,4, u - 2)(u + 5, u + 1,2u - l),

ef = (u + 2,2,4)(u e-‘ef=

+ 3, u - 1, l),

+ 3, u - 1, l),

(u + 2, u,4).

The connectedness of the set of 3-cycles in H is easily checked. Thus, Lemma 7 applies. 0 LEMMA 16. If r < k - 1 and H(r) induces at least A,,, on [0,2’u), H( r + 1) induces at least A2,+lo on [0,2’+‘u). Proof: We will restrict all permutations Lemma 14, H(r) contains t = (012)(2’u,2’u

then

to [0,2’+ ‘u). By hypothesis and

+ 1,2’u + 2).

By Lemma 10, since 2% = 2’(u - 1) + 2’ we have th(r+l)

=

(y+’

-

1,2’+1

-

2,3+1

-

3)(2’(u +2’

- 1) + 2’-

1,2’(u

- 1)

2,2’( u - 1) + 2’ - 3)

(if r = 1, replace 2’( u - 1) + 2’ - 3 by 2( u + 1) + 2’+’ - 1). Let g E H(r) fix the last 5 points appearing above in t”(‘+ ‘) and send 2’+ ’ - 1 to 2’+‘. Then tw+

y

tw+

I’)- ’ = (y+ 1,2’+ 1 - 1,2r+ 1 _ 3).

Conjugating by elements of H(r), and by h(r + l), we obtain enough 3-cycles to deduce the connectedness of the set of 3-cycles in the action of H(r + 1) on [0,2’+‘). •I LEMMA 17. If u > 3, then G* 3 A,. Proof. This follows from Lemmas 15 and 16 since H(k least A,, on [0, 2k-~u). •I LEMMA 18. If u = 3 and k & 4, then G* 2 A,,.

- 1) induces at

188

DIACONIS, GRAHAM, AND KANTOR

Proof: Restricting H(3) to [0,23 - 3), we obtain generated by the permutations b=

the subgroup of S,,

(024)(6810)(121416)(182022)(531)(1197)(171513)(2321

19);

c=(012)(345)(678)(91011)(121314)(151617)(181920)(212223); h(1):

2x @ 2x + 1;

h (2) : ( 4x4;r ;$4; h(3).

:‘2;

8x - 8x + 7, 8x+ 1 ++8x+6, * 8x+2*8x+5, 1 8x + 3 - 8x + 4.

A computer calculation shows that these generate A,, indeed b, c, and h(3) generate A,. (It may be of interest to observe that b, c, and h(2) generate M,, .) Now Lemma 16 applies. 0 Completion of Proof: We may assume that 2n is not a power of 2, and that 2n * 12,24. By Lemmas 17, 18, and 8, G* 3 A, and llyl 2 2”-‘. If n = 0 (mod4), then, by Lemma 6(a), c = A, and IGI d 2”-‘IA,,l. Thus, (d) holds. If n = 2 (mod 4), then, by Table III, c = S,. Also by Table III, 0 is odd while 0 is even. Let g E G* with g = 0. Then g0 -’ is an odd element of K. As in Lemma 9, the proof of Lemma 8 yields that llyl = 2”, as required. 0 3. SOME HISTORY OF THE PERFECT SHUFFLE There are early descriptions of the perfect shuffle in books on cheating at cards. The first description we can find is on p. 91 of the anonymously authored “Whole Art and Mystery of Modem Gaming,” Roberts, London, 1726. The earliest American reference to perfect shuffles we can locate is in J. H. Green’s book “An Exposure of the Arts and Miseries of Gambling,” James, Cincinnati, 1843. On p. 195 he described a method of cheating at the game of Faro which used the perfect shuffle, calling it “running in the cards.” Green remarked that the method was a recent invention. The perfect shuffle is currently called the Faro shuffle in magic circles. It is still widely used as a method of cheating at card games such as gin rummy and poker. A detailed description of several uses of the perfect shuffle for cheating at Faro can be found in the anonymously authored book “A Grand Expose of the Science of Gambling” Brady, New York, 1860. An illustrated description of the technique is on pp. 204-205 of J. N. Maskelyne’s book “Sharps and Flats,” Longmans Green and Co., London, 1894.

PERFECT

SHUFFLES

189

The shuffle was introduced to magicians in a brief note in C. T. Jordan’s classic “Thirty Card Mysteries,” published by the author in Pengrove, Calif., 1919. In a trick called the Full Hand, on pp. 17- 19, he used the principle that a 16 card packet out shuffled 4 times returned to its original order. He mentioned that 5 out shuffles suffice for 32 cards and that T. Nelson Downs, a famous American manipulator, could do similar things with 52 cards. It is through Downs that we have any record of a skillful early practioner of the perfect shuffle: Fred Black, a rancher from Thedford, Nebraska. Black worked out the mathematics of repeated out shuffles of 52 cards in some detail. Downs reported meetings with Black in 1924. These letters are reprinted in the magic journal, The Linking Ring, April (1971), 53-83 and May (1971), 67-68; 72-73. Dai Vernon knew Black and said that he used to practice shuffling on horseback. The charts Black worked out depicting some of the group structure of out shuffles appeared first in Hugard and Braue’s “Expert Card Technique,” Chap. 16. These charts record facts like: cards 18 and 35 repeatedly interchange during repeated out shuffles; the deck breaks up into groups: top, bottom, {18,35), 6 “belts” of 8 cards which are permuted among themselves, etc. The modem era for perfect shuffles in magic begins in 1957 when J. Russell Duck, a Pennsylvania policeman, published the basic central symmetry principle, calling it “stay-stack’, in the first issue (Feb. 1957) of the privately published journal Curdiste. At about the same time, Alex Elmsley, a computer specialist living in London, began to publish a series of tricks based on the perfect shuffle. In the privately published card magazine Ibidem (No. 11, Sept. 1957), he established the binary procedure for bringing the top card to any position (Lemma 2). In a series of articles in the English magic journal Pentugrum 11 (1957), he set out the basic mathematics for decks of general size, discovering in particular the importance of the order of 2 mod(2n f l), and the connection with Fermat’s little theorem. While many new tricks based on Faro shuffles have appeared in the magic literature in recent years, few new properties have emerged. (Our theorem, in a sense, explains this.) A scholarly manuscript by Ronald Wohl, a Swiss chemist, completed in the early 1960s has recently been published, in part, in Ibidem, No. 36. In two books, “Far0 Fantasy” and “More Faro Fantasy,” Paul Swinford discovered that with a deck of 2k cards, the shuffle sequence (in Lemma 2) that brings card x to position y also brings the card at y to position x. Swinford (private communication) also discovered the method for bringing a card at x to position y for 2k cards. There are a large number of recent articles on the Faro shuffle. Two excellent books by Edward Marlo, “The Faro Shuffle” and “Far0 Notes,” Ireland Magic Company, Chicago, are currently sold in magic shops.

190

DIACONIS,

GRAHAM,

AND

KANTOR

TABLE IV Cycle and Type for Out Shuffles of 64 Cards cydc

nv

(0)

(0)

(1,2,48,16,32)

(000001)

(3,6,12,24,48,33)

(000011)

(5.10,20,40,17,34)

(000101)

(7.14,28,56,49,35)

(000

(9.18,36)

(00 1)

(11,22,44,25,50,37)

(001011)

(13,26.52,41.19,38)

(001101)

(15.30,60,57,51,40’)

(001

(21.44)

(011)

(23,46,29,58,43)

(010111)

(27,54,45)

(011)

(31,62,61,59,55,47)

(lQOOO0)

(63)

(1)

1 11)

11 1)

There has been some mathematical work on perfect shuffles. P. Levy wrote a sequence of papers on them in 1940-1950. These papers (Levy [ 13- 181) are together in Vol. 6 of Levy’s collected works. Levy’s motivation for working on these shuffles is charmingly described in his autobiography [19, pp. 151-1531. Since Levy’s work seems unknown, a brief description is presented. Throughout, a deck of size 2n is assumed. We give results for out shuffles. Levy defined the type of a cycle in the cycle decomposition of an out shuffle as follows: suppose a card at position j, goes through positions j,, j2,..., j,, in successive out shuffles. The type of the cycle (j,, j,, . . . , j,) is a binary vector of length u with a zero in the i th position if and only if 0 Q ji G n. Table IV lists the cycle and type decomposition for a deck of 64 cards. Levy proved that all cycles have distinct types. When n = 2k he showed that all types occur if the following conventions are made: two types that differ by a cyclic shift are equivalent. Call a type imprimitiue if it is made up of repetitions of a single shorter type. (For 64 cards, (0 0 0 0 0 0) (0 0 1 0 0 1) (0 1 0 1 0 l)(l 1 0 1 1 0) (1 1 1 1 1 1) are imprimitive.)

PERFECT SHUFFLES

191

Only the shorter types appear for an imprimitive type. F. Leighton has pointed out that the type result just stated is equivalent to the representation of an out shuffle as a shift operating on Z,“, as in the proof of Lemma 4. A primitive type is “born” at k, if it occurs in a deck of k, cards and for no smaller deck. Levy showed that if a type is born at k,, then it appears in a deck of size k if and only if k = k, + 2(k, - l)j, j = 0, 1,2,. . . . For example, since a 2-cycle appears for the first time with k, = 4 cards it follows that decks of size k = 4 + 6j have 2-cycles. Levy proved that for a fixed k, all the new born types are of the same length u. This u is the order of 2 (mod k - 1). Levy gave a number of other results. Golomb [9] considered the group generated by out shuffles and cuts. He showed that these operations generate all permutations. He also gave results for a deck of size 2n - 1. One implication of his results is that in and out shuffles of an odd-sized deck generate a very small group. (Here, in and out shuffles correspond to the two ways of cutting a deck into two parts of size n and n - 1. The out shuffle leaves the top card on top, while the in shuffle leaves the bottom card on bottom.) THEOREM. The order of the shuffle group for a deck of 2n - 1 cards is (2n - 1) X order of 2 mod(2n - 1). Proof: Let c be the operation of cutting the top card to the bottom. It is easy to see that an out shuffle followed by c is the same as an in shuffle. It follows that (0,I) = (0, c). The order of the latter group was shown to be (2n - 1) X order of 2 (mod(2n - 1)) by Golomb. 0

In fact, if the cards are labeled using [0,2n - l), then x0 = 2x (mod 2n - l), xl = 2x + 1 (mod2n - l), and the cut c is given by xc = x + 1 (mod 2n - 1). The order of (0, c) is easily found using the identity co = c2 observed by Golomb. In terms of 0 and I = Oc this asserts that I0 = to-1 ’ I-’ . The magical properties of perfect shuffles of odd-sized decks are discussed at length in Gardner [8, Chap. lo].

4. APPLICATIONS OF PERFECT SHUFFLES TO PARALLEL PROCESSING ALGORITHMS

Both types of perfect shuffles and their combinations have been applied to computer algorithms. Some references are Stone [25], Schwartz [23], and Chen et al. [4]. For a simple example, due to Stone, consider computing the transpose of a 2” X 2m matrix. Suppose that the matrix is stored in a 22m linear array in row major order. For a 4 x 4 matrix this is It is easy to verify that aooao1a02a03a10a11a12L113c120a21a22a23a333.

192

DIACONIS,

GRAHAM,

AND KANTOR

after m out shuffles the array is in column major order, and so transposed. In the 4 X 4 example this is a 00 a 10 a 20 a 30 a II a 21 a 31 a 12 a 32 a 03 a 13 a 23 a 33’ Out shuffles are performed by network connection patterns like the example in Fig. 1 in which 2 sets of 8 registers are connected. Often an out shuffle connection is combined with an array of simple processors (Fig. 2). If the processors P take two input numbers and output their sum (Fig. 3) then, after m iterations of a network with 2” registers, all registers will contain the sum a, + a, + - . - + a2,,, _ ,. If the processors output other 0

0

1

4

2

1

3

5

4

2

5

6

6

3

7

7 FIGURE I

FIGURE 2

FIGURE 3

PERFECT SHUFFLES

193

suitably chosenlinear combinations the result of m iterations is the discrete Fourier transform Zujwjk. In a more complex application the processors output the sorted pair (ai, uj). This yields an algorithm that sorts 2” numbers in 0( m2) iterations. See Brock et al. [3] for further discussion. It should be emphasized that these applications mostly amount to the “out” part of the proof of Lemma 4. Namely, when 0 is used on a deck of size 2” it cyclically shifts the digits in the binary expansion of each integer in [0,2”). As a simple new example, we mention the fact that an array of 2” numbers is brought into reverse order by m in shuffles. Other length arrays may also be reversed; for length 52, 26 in shuffles suffice. In general, an array of length 2n reverses after j in shuffles, where i is the smallest exponent such that 2j = - l(mod2n + 1). In shuffles cannot always be called upon to reverse an array of length 2n. For example, if 2n = 2” - 2, the order of 2 (mod2n + 1) is m and 2j g - 1 (mod2n + 1) for any j. A simple corollary of the main theorem is that an array of length 2n can be reversed by some combination of in and out shuffles if and only if n = 0, 1, or 2 (mod4). Indeed, if z denotes the permutation reversing the position of 2n symbols, sgn z = (- l)“, sgn Z = 1, now apply Lemma 5 and the theorem. As an example, if 2n = 10, the sequence IO 0 IO 0 0 0 0 yields z. We do not know a simple algorithm for determining a minimal length sequence for general n.

5. GENERALIZATIONS AND VARIATIONS In and out shuffles are related to the so-called “milk” and “Monge” shuffles. The milk shuffle can be described as a permutation on m cards labeled 0, 1, . . . , m - 1 as follows. The card labeled j (and initially in position j) is moved to position 125’1, where 1x1 denotes the unique y, 0 G y G m, such that y = f x (mod 2m - 1). Thus, after one milk shuffle, the deck now has the order 0, m - 1, 1, m - 2,2,. . . . This permutation is easily performed on a deck of cards. Remove, or “milk,” the cards at top and bottom simultaneously and place them on the table. Then milk off the second pair from the top and bottom and place them on top of the first removed pair. Continue this process until no cards remain. Basic properties of the milk shuffle were given by Levy in the papers cited in the bibliography. Levy [18] showed that the results he proved for milk shuffles had easy translations into corresponding results for out shuffles. A useful connection between the two shuffles is the following observation due to John Conway. The permutation of the pairs 3 in an out shuffle of 2m cards is just a milk shuffle of the m symbols X0, Z,, . . . , Tm _ ,. The inverse of the milk shuffle was actually analyzed by Monge in 1773 (see Ball and Coxeter [ 11). In fact, there are two types of “Mange” shuffles, which we will call the

194

DIACONIS,

GRAHAM,

AND

KANTOR

“up” shuffle and the “down” shuffle. For the up shuffle, successive cards are removed from the top of the deck and placed alternatively on the top and the bottom of the new stack (with the second card being placed on top of the first). For the down shuffle, the same procedure is followed except that the second card is placed below the first. In particular the order of a milk shuffle of m cards is the order of an out shuffle of 2m cards: the order of 2 (mod 2m - 1). Symbolically, for a deck of m cards, the up shuffle sends a card originally in position j to position [m/2] + (- l)‘[(i + 1)/2]. Note that a down shuffle is actually just an up shuffle followed by a “reversal” shuffle z, i.e., z(j) = m - 1 - j. It follows from what we have noted that the group (u, d) of-- permutations generated by up and down shuffles on m cards is exactly (I, 0) acting on (pairs of) 2m cards. In particular, since (u, d) = (u, z), it follows that for m = 2 (mod 4) (u, z) z S,,, while for m = 12, (u, z) Z M,*. Borcherds et al. [2] have used this fact in order to relate two different bases for the Leech lattice. Levy also mentioned a curious connection between the milk shuffle and the down and under shuffle. This shuffle successively places the top card on the table, the next card under the deck, the next card on (top of the card on) the table, and so on. Let E be the set of integers with the property that the milk shuffle and the down and under shuffle have the same cycle structure (and in particular the same order). Levy showed that n E E if and only if 2n - 1 divides a number of the form 2’ + 1. For example, a milk shuffle with 5 cards labeled 01234 gives the permutation 0 0

1 2

2 4

3 3

4 1.

A down and under shuffle gives the permutation 0 0

1 4

2 1

3 3

4 2.

Both permutations have one 3-cycle and two fixed points, and 2n - 1 = 9 which divides 23 + 1. A. M. Odlyzko has shown that E is small in the sense of density: the number of elements in E smaller than x is asymptotic to C~/m3 x) ‘I3 for an explicit constant c. The down and under shuffle is thoroughly studied under the name of the Josephus permutation (see Herstein and Kaplansky [ 111). The milk shuffle is described and applied in early books on card cheating. For example, the anonymously authored book “Whole Art and Mystery of Modern Gaming” (London, 1726) contains a description of several methods of cheating at the card game of Faro that make use of milk shuffles. A simple way to achieve an inverse in or out shuffle is to deal a deck of cards into two face-up piles alternatively. Place one pile on the other, and

PERFECT SHUFFLES

195

turn the deck face down. This suggests a generalized perfect out shuffle: for a deck consisting of a . b cards, deal a face-up piles and gather the piles so that the original top card remains on top. This shuffling operation arises naturally in circuit applications such as the final shuffling in the Cooley-Tukey algorithms for the discrete Fourier transform. Davio [6] contains a nice discussion of this and other examples. One application involves the noncommutativity of tensor products. If the permutation matrix of the generalized out shuffle is denoted S,, b and if m, and m, are (r,, c,) and (ra, cO) matrices, then Davio [6] showed that

Morris and Hartwig [22] have determined properties of generalized out shuffles and generalized out shuffles and cuts; they also proved a special case of the above formula (when r, = c, and r0 = c,,). We observe that it is also possible to define generalized in shuffles by picking up the piles in the opposite order. Both generalized in and out shuffles preserve central symmetry; it would be interesting to know what group these generate. Going further, if a . b cards are dealt into a piles with b cards in each pile, there are a! possible ways of picking up the piles, and this leads to more questions of the same sort.

ACKNOWLEDGMENTS This paper makes heavy use of computational results generated by Eric Hamilton in a course at Stanford taught by Don Knuth. The calculations leading to M,, and a crucial computer check required in the proof of the main theorem were carried out by Lyle Ramshaw at Xerox PARC. Rob Calderbank and Neil Sloane provided the first corroboration that for n = 12, c = M,,. A. M. Odlyzko produced the difficult estimate in Section 5. We thank them all for their help.

REFERENCES 1. W. W. R. BALL AND H. M. S. COXETER, “Mathematical Recreations and Essays,” 12th ed., University of Toronto Press, Buffalo, 1974. 2. R. BORCHERDS,J. H. CONWAY, L. QUEEN, AND N. J. A. SLOANE, “A Monster Lie Algebra?” Technical Report, Bell Laboratories, Murray Hill, N.J., 1982. 3. H. K. BROCR, B. J. BROOKS, AND F. SULLIVAN, Diamond, a sorting method for vector machines, BIT 21(1981), 142-152. 4. P. Y. CHEN, D. H. LAWRIE, P. C. YEW, AND D. A. PODERA, Interconnection networks using shuffles, Computer, December (198 I), 55-64. 5. H. M. S. COVETER,“Regular Polytopes,” Methuen, London, 1948. 6. M. DA~IO, Kronecker products and shuffle algebra, IEEE Trans. Comput. C-30 (1981) 116-125.

196

DIACONIS,

GRAHAM,

AND KANTOR

7. M. FURST, J. HOPCROR, AND E. Luxs, Polynomial-time algorithms for permutation groups, in “Proc. 21st FOCS,” pp. 36-41, 1980. 8. M. GARDNER, “Mathematical Carnival,” Knopf, New York, 1975. 9. S. W. GOLOMB, Permutations by cutting and shuffling, SIAM Rev. 3 (1961), 293-297. IO. R. E. HARTWIG AND S. B. MORRIS, The universal flip matrix and the generalized Faro shuffle, Pacific J. Math. 58 (1975), 445-455. Il. I. N. fhCXEIN AND I. KAPLANSKY, “Matters Mathematical,” Harper & Row, New York, 1974. 12. D. KLEITMAN, F. LEIGHTON, M. LEPLEY, AND G. MILLER, “New Layouts for the Shuffle-Exchange Graph,” Technical report, Applied Mathematics Department, MIT., Cambridge, Mass., 1980. 13. P. LEW, Etude dune classe de permutations, C. R. Acad. Sci. 227 (1948), 422-423. 14. P. LEW, Etude dune nouvelle classe de permutations, C. R. Acad. Sci., 227 (1948), 578-579. 15. P. LEW, Sur dew classes de permutations, C. R. Acad. Sci. 228 (1?49), 1089-1090. 16. P. LEW, Un problem de la theorie des permutations, in “Conf. Ecole Polyt.,” May 24, 1949. 17. P. LEW, Sur une classe remarquable de permutations, Bull. Acad. Roy. Belgique 2 (1949). 361-377. 18. P. LEW, Sur quelque classes de permutations, Compositio Math. 8 (1950), l-48. 19. P. LEW, “Quelques aspects de la pensee d’un mathematician,” Blanchard, Paris, 1970. 20. S. B. MORRIS, “Permutations by Cutting and Shuffling-A Generalization to Q-Dimensions,” Ph.D. dissertation, Department of Mathematics, Duke University. Published under the same title by Micky Hades, Box 476, Calgary, Alberta, Canada. 21. S. B. MORRIS, The basic mathematics of the Fare shuffle, Pi Mu Epsilon J. 6 (1975), 85-92. 22. S. B. Mows AND R. E. HARTWIG, The generalized Faro shuffle, Discrete Math. 15 (1976), 333-346. 23. J. T. SCHWARTZ, Ultracomputers, ACM Trans. Program. Language Systems 2 (1980), 484-521. 24. C. C. SIMS, Computational methods in the study of permutation groups, in “Computational Problems in Abstract Algebra,” Proc. Conf., Oxford, 1967 (John Leech, Ed.), pp. 169- 183, Pergamon, Oxford, 1970. 25. H. S. STONE, Parallel processing with the perfect shuffle, IEEE Trans. Comput. 2 (1971), 153-161. 26. J. V. USPENSKY AND M. A. HEASLET, “Elementary Number Theory,” McGraw-Hill, New York. 1939.